The probability of dry weather (D) = 1⁄2, rain (R) = 1⁄3 and snow (S) = 1⁄6 If it is dry the probability that someone will arrive on time is 4
/5.
If it rains the probability that someone will arrive on time is 2
/5.
If it snows the probability that someone will arrive on time is 1
/10.

Calculate the probability that somebody is late.

Explanation:

Define the following events

We are given the following probabilities

Then we're further told that "if it is dry, then the probability someone will arrive on time is 4/5". We can condense that into the notation

P(T given D) = 4/5

The "given D" means that we know 100% that event D has occurred. So we're seeing how event T behaves because of D occurring. In other words P(T given D) is the same as saying "What is P(T) when we know D has happened?"

The instructions also mention that "If it rains, the probability someone is on time is 2/5" tells us we have the notation

P(T given R) = 2/5

And also, the probability someone is on time if it is snowy is

P(T given S) = 1/10

due to it saying "If it snows, the probability that someone will arrive on time is 1/10".

To recap so far, your teacher told you these three conditional probabilities

They are conditional due to the "given" as part of the probability. They depend on the given event happening.

Subtract each fraction from 1 and we end up with these complementary probabilities

I'm using the idea that P(A)+P(B) = 1, where A and B are complementary events. One or the other event must happen. Either you are on time, or you are late.

Recall the conditional probability formula is

P(A given B) = P(A and B)/P(B)

which rearranges to

P(A and B) = P(A given B)*P(B)

We'll use this idea to get the following

This means,

P(L) = P(L and D) + P(L and R) + P(L and S) law of total probability

P(L) = 1/10 + 1/5 + 3/20

P(L) = 2/20 + 4/20 + 3/20

P(L) = (2+4+3)/20

P(L) = 9/20 is the probability someone is late

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step-by-step explanation: